As we can see, on the one hand, interpreting shells as Fourier modes is not fully satisfactory. On the other hand, it is tempting to interpret shells as wavelet coefficients in a dyadic decomposition of velocity over a tree. Such a decomposition can been shown to be orthonormal for square-integrable functions, and possesses a reconstruction formula in physical space [1]}, that has been used in a turbulent context [2]}, [3]} in order to synthesize random fields able to reproduce aforementioned statistical properties (Eqs. REF  and REF ). Doing so, inverting this orthonormal decomposition in order to get the dynamics in the physical space requires to link these shells, or wavelet coefficients, both in scale and in space. This interpretation of shell models, much more complete than only considering interactions through scales, has been already explored in the literature [4]}, [5]}. Nevertheless, an analytical derivation of the statistical properties of such shells when the dynamics is forced by an external large-scale forcing remains difficult since further relations between these coefficients in space must be prescribed, in order
to guarantee, for instance, spatial homogeneity of the velocity field in physical space, i.e. that the underlying probability law is invariant by translation. Designing such an interaction between the shells is not obvious and barely discussed. Up to now, we are not aware of such a dyadic model over a tree able to reproduce the rough behaviors depicted by the behaviors of structure functions at small scales (Eqs. REF  and REF ) in a statistically stationary and homogeneous framework.
It is also worth mentioning the approach of the so-called Leith [6]}, [7]} and EDQNM [8]}, [9]} models in directly proposing a PDE for the energy spectrum, based on the phenomenology of turbulence. Nevertheless, these models do not address fundamental statistical features of the underlying velocity field, such as homogeneity, stationarity and intermittency.
